In our calculations for band 1 we choose \(\mu _1 = 30\)  meV, which is in the range of the Fermi energies in recent multiband superconductors [1]}. The chemical potential relative to the lower edge of band 2 (\(\mu _2\) ) is varied in order to consider different values of the ratio \(v_{F2}/v_{F1}\) . We also use the cut-off energy \(\hbar \omega _D = 15\)  meV and the dimensionless intraband coupling \(g_{11}N_1 = 0.3\) , with \(N_i\)  the density of states (these values are typical of the conventional superconductors [2]}). For band 2 we take \(g_{22}=0.8g_{11}\) . The two values \(g_{12}=0.3g_{11}\)  and \(0.9g_{11}\)  are considered for the pair-exchange (interband) coupling to check both the weak- and strong-coupling regimes of the intraband interactions. We note that most of multiband superconductors exhibit weak pair-exchange couplings but there are also examples with relatively strong intraband interactions like MgB\(_{2}\) , see e.g. table II of Ref. Vagov2016 and also Refs. Golubov2002,Singh2010,Khasanov2010,Kim2011.
